§2.4 Derivatives of other trigonometric functions

Introduction

Derivatives of the cotangent, secant, and cosecant functions

Since $\tan(x) = \frac{\sin(x)}{\cos(x)}$, the domain of the tangent function consists of

Since $\csc(x) = \frac{1}{\sin(x)}$, the domain of cosecant is all real $x$ except $x = k\pi$. The derivative $-\csc(x)\cot(x)$ then

Which of the following is the derivative of the secant function?

Which of the following is the derivative of the cosecant function?

For $f(x) = 5\sec(x) - 2\csc(x)$, compute the tangent slope $f'(\frac{\pi}{3})$ rounded to three decimals.

Let $p(z) = z^2\sec(z) - z\cot(z)$. Which structure does $p'(z)$ have?

Let $h(t) = \frac{\tan(t)}{t^2+1} - 2e^t\cos(t)$. Which structure does $h'(t)$ have?

For $g(r) = \frac{r\sec(r)}{5^r}$, the quotient and product rules give $g'(0) = \frac{5^0[\sec(0) + 0] - 0}{5^0}$. Compute $g'(0)$.

A damped oscillator has displacement $s(t) = \frac{15\sin(t)}{e^t}$ inches at time $t$ seconds. Compute the velocity $s'(2)$ rounded to two decimals.

The functions $\sec(x)$ and its derivative $\sec(x)\tan(x)$ have the same domain.